#60 ⟨a, b | aa=b, ab=b⟩
Contents
- Properties
- Commutative structure
- Finite structure
- Staircase diagram
- Cayley table
- Right Cayley graph
- Rewriting system
- Same cardinality
- Isomorphic instances
- Sum of relation sides is 6
- Isomorphic to ℕ(3 = 2)
- Finite commutative monoid with 3 elements
- Not cancellative, because multiplication by a is not injective:
-
a ⋅ a2 = a2 and a ⋅ a = a2, however a2 ≠ a
- Gröbner basis: ⟨a, b | a3=a2, b=a2⟩
- Cancellative quotient is isomorphic to ℤ1
- Enveloping group is isomorphic to ℤ1
- Group of units is isomorphic to ℤ1
- 2 Archimedian components:
The zero element z satisfies zy = yz = z for all y.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.
- Zero element: a2
- 1 non-trivial idempotent:
- Order of generators:
- a: index 2, period 1
- b: index 1, period 1
- Histogram:
| index 1, period 1 | 1 element | a2 |
| index 2, period 1 | 1 element | a |
Idempotents are shown in bold.
|
1 | a | a2 |
|---|
| 1 | 1 | a | a2 |
|---|
| a | a | a2 | a2 |
|---|
| a2 | a2 | a2 | a2 |
|---|
Idempotents are shown in bold.
- Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1
- Certificate: derivations of all rewriting rules from the defining relations.
# ab:aa=b,ab=b a/b - -
aaa=aa
b=aa
4 unique, 2184 total
The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.
23 total
| Σ | # | Presentation | Mapping |
|---|
| 7 | 248 | ⟨a, b | aa=b, aaa=b⟩ | φ(a) = a, φ(b) = aa |
| 7 | 277 | ⟨a, b | aa=b, ab=aa⟩ | φ(a) = a, φ(b) = aa |
| 7 | 281 | ⟨a, b | ab=a, bb=ab⟩ | φ(a) = aa, φ(b) = a |
| 8 | 972 | ⟨a, b | aa=b, aaa=aa⟩ | φ(a) = a, φ(b) = aa |
| 8 | 1007 | ⟨a, b | ab=a, abb=bb⟩ | φ(a) = aa, φ(b) = a |
| 9 | 1613 | ⟨a, b | aab=aa, abb=b⟩ | φ(a) = a, φ(b) = aa |
| 9 | 3103 | ⟨a, b | ab=a, abbb=bb⟩ | φ(a) = aa, φ(b) = a |
| 10 | 5023 | ⟨a, b | aaa=aa, aaaa=b⟩ | φ(a) = a, φ(b) = aa |
| 10 | 5109 | ⟨a, b | aab=aa, abbb=b⟩ | φ(a) = a, φ(b) = aa |
| 10 | 9047 | ⟨a, b | ab=a, abbbb=bb⟩ | φ(a) = aa, φ(b) = a |
| 11 | 12217 | ⟨a, b | aaab=aa, aabb=b⟩ | φ(a) = a, φ(b) = aa |
| 11 | 12221 | ⟨a, b | aaab=aa, abab=b⟩ | φ(a) = a, φ(b) = aa |
| 11 | 12223 | ⟨a, b | aaab=aa, abba=b⟩ | φ(a) = a, φ(b) = aa |
| 11 | 12225 | ⟨a, b | aaab=aa, abbb=b⟩ | φ(a) = a, φ(b) = aa |
| 11 | 12333 | ⟨a, b | aaba=aa, abba=b⟩ | φ(a) = a, φ(b) = aa |
| 11 | 15402 | ⟨a, b | aaa=aa, aaaaa=b⟩ | φ(a) = a, φ(b) = aa |
| 11 | 15576 | ⟨a, b | aab=aa, abbbb=b⟩ | φ(a) = a, φ(b) = aa |
| 11 | 19407 | ⟨a, b | aab=a, aabbb=bb⟩ | φ(a) = aa, φ(b) = a |
| 11 | 19423 | ⟨a, b | aab=a, ababb=bb⟩ | φ(a) = aa, φ(b) = a |
| 11 | 19431 | ⟨a, b | aab=a, abbab=bb⟩ | φ(a) = aa, φ(b) = a |
| 11 | 19435 | ⟨a, b | aab=a, abbba=bb⟩ | φ(a) = aa, φ(b) = a |
| 11 | 19439 | ⟨a, b | aab=a, abbbb=bb⟩ | φ(a) = aa, φ(b) = a |
| 11 | 24987 | ⟨a, b | ab=a, abbbbb=bb⟩ | φ(a) = aa, φ(b) = a |